# Radiative Transfer In A Gray Atmosphere

Now we have discussed the mechanisms by which gases may absorb IR radiation we will consider how the thermal emission spectra of planets are generated. Figure 6.6. Near-IR transmission of tropospheric gases for a solar composition path with p = 0.3 bar, T = 127 K, and pathlength = 10 km.

When considering radiation that is emitted to space at angles reasonably close to the vertical, it is a very good approximation to consider the atmosphere as being effectively plane-parallel. This approximation breaks down at very high emission angles such as when the limb of the planet is observed (but even then the radiative transfer equations turn out to be rather similar). Consider such a plane-parallel atmosphere as shown in Figure 6.7. Lambert's law of absorption states that at wavenumber v the absorption of radiation of spectral radiance Iv (typically measured in units of Wcm-2 sr-1 (cm-1)-1) traveling at a zenith angle 6 to the vertical at altitude z through a path of vertical thickness dz (meters), mean number density n (molecules per cubic meter) Figure 6.7. Radiative transfer in a gray plane-parallel atmosphere.

containing material with absorption coefficient kp (square meters per molecule) is given by dIp = —Ikyn dz/cos d. (6.32)

In other words the layer element, of thickness dz, has absorption along the slant path equal to a = kpn dz/p, where p = cos d. Consider a thick slab of atmosphere between levels z0 and z,, with spectral radiance Ipo incident upwards at z0 at an angle d. This radiation will be attenuated before reaching z, and thus upward radiation at the top of the atmosphere from the base is given by